-Functions on Quasimetric Spaces and Fixed Points for Multivalued Maps

Abstract

We discuss several properties of -functions in the sense of Al-Homidan et al.. In particular, we prove that the partial metric induced by any weighted quasipseudometric space is a -function and show that both the Sorgenfrey line and the Kofner plane provide significant examples of quasimetric spaces for which the associated supremum metric is a -function. In this context we also obtain some fixed point results for multivalued maps by using Bianchini-Grandolfi gauge functions.

1. Introduction and Preliminaries

Kada et al. introduced in [1] the concept of -distance on a metric space and extended the Caristi-Kirk fixed point theorem [2], the Ekeland variation principle [3] and the nonconvex minimization theorem [4], for -distances. Recently, Al-Homidan et al. introduced in [5] the notion of -function on a quasimetric space and then successfully obtained a Caristi-Kirk-type fixed point theorem,a Takahashi minimization theorem, an equilibrium version of Ekeland-type variational principle, and a version of Nadler's fixed point theorem for a - function on a complete quasimetric space, generalizing in this way, among others, the main results of [1] because every -distance is, in fact, a -function. This interesting approach has been continued by Hussain et al. [6], and by Latif and Al-Mezel [7], respectively. In particular, the authors of [7] have obtained a nice Rakotch-type theorem for -functions on complete quasimetric spaces.

In Section 2 of this paper, we generalize the basic theory of -functions to quasipseudometric spaces. Our approach is motivated, in part, by the fact that in many applications to Domain Theory, Complexity Analysis, Computer Science and Asymmetric Functional Analysis, quasipseudometric spaces (in particular, weightable quasipseudometric spaces and their equivalent partial metric spaces) rather than quasimetric spaces, play a crucial role (cf. [8–23], etc.). In particular, we prove that for every weighted quasipseudometric space the induced partial metric is a -function. We also show that the Sorgenfrey line and the Kofner plane provide interesting examples of quasimetric spaces for which the associated supremum metric is a -function. Finally, Section 3 is devoted to present a new fixed point theorem for -functions and multivalued maps on quasipseudometric spaces, by using Bianchini-Grandolfi gauge functions in the sense of [24]. Our result generalizes and improves, in several ways, well-known fixed point theorems.

Throughout this paper the letter and will denote the set of positive integer numbers and the set of nonnegative integer numbers, respectively.

By a quasipseudometric on a set , we mean a function such that for all ,

(i),

(ii).

A quasipseudometric on that satisfies the stronger condition

(i′)

is called a quasimetric on .

We remark that in the last years several authors used the term "quasimetric" to refer to a quasipseudometric and the term " quasimetric" to refer to a quasimetric in the above sense.

In the following we will simply write qpm instead of quasipseudometric if no confusion arises.

A qpm space is a pair such that is a set and is a qpm on . If is a quasimetric on , the pair is then called a quasimetric space.

Given a qpm on a set , the function defined by , is also a qpm on , called the conjugate of , and the function defined by is a metric on , called the supremum metric associated to .

Thus, every qpm on induces, in a natural way, three topologies denoted by , and , respectively, and defined as follows.

(i) is the topology on which has as a base the family of -open balls , where , for all and .

(ii) is the topology on which has as a base the family of -open balls , where , for all and .

(iii) is the topology on induced by the metric .

Note that if is a quasimetric on , then is also a quasimetric, and and are topologies on .

Note also that a sequence in a qpm space is -convergent (resp., -convergent) to if and only if (resp., .

It is well known (see, for instance, [26, 27]) that there exists many different notions of completeness for quasimetric spaces. In our context we will use the following notion.

A qpm space is said to be complete if every Cauchy sequence is -convergent, where a sequence is called Cauchy if for each there exists such that whenever .

In this case, we say that is a complete qpm on .

2. -Functions on qpm-Spaces

We start this section by giving the main concept of this paper, which was introduced in [5] for quasimetric spaces.

Definition 2.1.

A -function on a qpm space is a function satisfying the following conditions:

(Q1), for all ,

(Q2) if , and is a sequence in that -converges to a point and satisfies , for all , then ,

(Q3) for each there exists such that and imply .

If is a metric space and satisfies conditions (Q1) and (Q3) above and the following condition:

(Q2′) is lower semicontinuous for all , then is called a w-distance on (cf. [1]).

Clearly is a -distance on whenever is a metric on .

However, the situation is very different in the quasimetric case. Indeed, it is obvious that if is a qpm space, then satisfies conditions (Q1) and (Q2), whereas Example 3.2 of [5] shows that there exists a qpm space such that does not satisfy condition (Q3), and hence it is not a Q-function on . In this direction, we next present some positive results.

Lemma 2.2.

Let q be a Q-function on a qpm space . Then, for each , there exists such that and imply .

Proof.

By condition (Q3), . Interchanging and , it follows that , so .

Proposition 2.3.

Let be a qpm space. If is a Q-function on , then , and hence, is a metrizable topology on .

Proof.

Let be a sequence in which is -convergent to some . Then, by Lemma 2.2, . We conclude that .

Remark 2.4.

It follows from Proposition 2.3 that many paradigmatic quasimetrizable topological spaces , as the Sorgenfrey line, the Michael line, the Niemytzki plane and the Kofner plane (see [25]), do not admit any compatible quasimetric which is a -function on .

In the sequel, we show that, nevertheless, it is possible to construct an easy but, in several cases, useful -function on any quasimetric space, as well as a suitable -functions on any weightable qpm space.

Recall that the discrete metric on a set is the metric on defined as , for all , and , for all with .

Proposition 2.5.

Let be a quasimetric space. Then, the discrete metric on is a -function on .

Proof.

Since is a metric it obviously satisfies condition (Q1) of Definition 2.1.

Now suppose that is a sequence in that -converges to some , and let and such that , for all . If , then . If , we deduce that , for all . Since , it follows that , so , and thus . Hence, condition (Q2) is also satisfied.

Finally, satisfies condition (Q3) taking for every

Example 2.6.

On the set of real numbers define as if , and if . Then, is a quasimetric on and the topological space is the celebrated Sorgenfrey line. Since is the discrete metric on , it follows from Proposition 2.5 that is a -function on .

Example 2.7.

The quasimetric on the plane , constructed in Example 7.7 of [25], verifies that is the so-called Kofner plane and that is the discrete metric on , so, by Proposition 2.5, is a -function on .

Matthews introduced in [14] the notion of a weightable qpm space (under the name of a "weightable quasimetric space"), and its equivalent partial metric space, as a part of the study of denotational semantics of dataflow networks.

A qpm space is called weightable if there exists a function such that for all . In this case, we say that is a weightable qpm on . The function is said to be a weighting function for and the triple is called a weighted qpm space.

A partial metric on a set is a function such that, for all :

(i),

(ii),

(iii),

(iv).

A partial metric space is a pair such that is a set and is a partial metric on .

Each partial metric on induces a topology on which has as a base the family of open -balls , where , for all and .

The precise relationship between partial metric spaces and weightable qpm spaces is provided in the next result.

where is a nondecreasing function satisfying , for all ( denotes the nth iterate of ). Then, has a unique fixed point.

A function satisfying the conditions of the preceding theorem is called a Bianchini-Grandolfi gauge function (cf [24, 30]).

It is easy to check (see [30, Page 8]) that if is a Bianchini-Grandolfi gauge function, then , for all , and hence .

Our next result generalizes Bianchini-Grandolfi's theorem for Q-functions on complete qpm spaces.

Theorem 3.3.

Let be a complete qpm space, q a Q-function on , and a multivalued map such that for each and , there is satisfying

(3.3)

where is a Bianchini-Grandolfi gauge function. Then, there exists such that and .

Proof.

Fix and let . By hypothesis, there exists such that . Following this process, we obtain a sequence with and , for all . Therefore

(3.4)

for all .

Now, choose . Let for which condition (Q3) is satisfied. We will show that there is such that whenever .

Indeed, if , then and thus , for all , so, by condition (Q1), whenever .

If , , so there is such that

(3.5)

Then, for , we have

(3.6)

In particular, and whenever , so, by Lemma 2.2, whenever .

We have proved that is a Cauchy sequence in (in fact, it is a Cauchy sequence in the metric space . Since is complete there exists such that .

Next, we show that .

To this end, we first prove that . Indeed, choose . Fix . Since whenever , it follows from condition (Q2) that whenever .

Now for each take such that

(3.7)

If , it follows that . Otherwise we obtain .

Hence, , and by Lemma 2.2,

(3.8)

Therefore, .

It remains to prove that .

Since , we can construct a sequence in such that , and

(3.9)

Since , it follows that , and thus . So, by Lemma 2.2, is a Cauchy sequence in (in fact, it is a Cauchy sequence in . Let such that . Given , there is such that , for all . By applying condition (Q2), we deduce that , so . Since , it follows from condition (Q1) that . Therefore, , for all , by condition (Q3). We conclude that , and thus .

The next example illustrates Theorem 3.3.

Example 3.4.

Let and let be the qpm on given by . It is well known that is weightable with weighting function given by , for all . Let be partial metric induced by . Then, is a -function on by Proposition 2.10. Note also that, by Theorem 2.8 (a),

(3.10)

for all . Moreover is clearly complete because is the Euclidean metric on and thus is a compact metric space.

Now define by

(3.11)

for all . Note that because the nonempty -closed subsets of are the intervals of the form , .

Let be such that , for all , and , for all . We wish to show that is a Bianchini-Grandolfi gauge function.

It is clear that is nondecreasing.

Moreover, , for all . Indeed, if we have whenever , while for , we have so,

(3.12)

and following this process we deduce the known fact that , for all . We have shown that is a Bianchini-Grandolfi gauge function.

Finally, for each and , there exists such that . Choose . Then and

(3.13)

If , then , and thus .

We have checked that conditions of Theorem 3.3 are fulfilled, and hence, there is with . In fact is the only point of satisfying and (actually . The following consequence of Theorem 3.3, which is also illustrated by Example 3.4, improves and generalizes in several directions the Banach Contraction Principle for partial metric spaces obtained in Theorem 5.3 of [14].

Corollary 3.5.

Let be a partial metric space such that the induced weightable qpm is complete and let be a multivalued map such that for each and , there is satisfying

(3.14)

where is a Bianchini-Grandolfi gauge function. Then, there exists such that and .

Proof.

Since (see Theorem 2.8), we deduce from Proposition 2.10 that is a -function for the complete (weightable) qpm space . The conclusion follows from Theorem 3.3.

Observe that if is a nondecreasing function such that , for all , then the function given by , is a Bianchini-Grandolfi gauge function (compare [31, Proposition 8]). Therefore, the following variant of Theorem 3.1, which improves Corollary 2.4 of [7], is now a consequence of Theorem 3.3.

Corollary 3.6.

Let be a complete qpm space. Then, for each generalized q-contractive multivalued map with q nondecreasing, there exists such that and .

Remark 3.7.

The proof of Theorem 3.3 shows that the condition that is complete can be replaced by the more general condition that every Cauchy sequence in the metric space is -convergent.

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Acknowledgments

The authors thank one of the reviewers for suggesting the inclusion of a concrete example to which Theorem 3.3 applies. They acknowledge the support of the Spanish Ministry of Science and Innovation, Grant no. MTM2009-12872-C02-01.